Recently, because of enhancement of an interest in environmental problems, fuel saving is intensely required for automobiles, and excellent fuel saving is required also for polymer compositions used in automobile tires. As the polymer compositions for automobile tires, use is made of polymer compositions containing a conjugated diene polymer such as polybutadiene, butadiene-styrene copolymer, and fillers such as carbon black, silica. For example, known is a polymer composition using, as the conjugated diene polymer, a polymer obtained by modification with a tin halide compound of a polymer prepared by copolymerizing butadiene and styrene using an alkyllithium as a polymerization initiator (for example, JP-A-60-255838, U.S. Pat. No. 4,742,124).
Further, there are suggested a polymer composition using, as the conjugated diene polymer, a polymer obtained by modification with an acrylamide having a dialkylamino group of a polymer prepared by copolymerizing butadiene and styrene using an alkyllithium as a polymerization initiator (for example, JP-A-1-217047, U.S. Pat. No. 5,189,109), a polymer composition using, as the conjugated diene polymer, a polymer obtained by modification with an alkoxysilane having a dialkylamino group of a polymer prepared by polymerizing butadiene or copolymerizing butadiene and styrene using an alkyllithium as a polymerization initiator (for example, JP-A-63-186748, U.S. Pat. No. 4,957,976, JP-A-2005-290355, US 2005/0203251 A1), and the like, as polymer compositions excellent in fuel saving.